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4*sin(t)^(2)
The derivative of the function/ 4*sin(t)^(2)

Derivative of 4*sin(t)^(2)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
     2   
4*sin (t)
4sin⁡2(t)4 \sin^{2}{\left(t \right)} 
d /     2   \
--\4*sin (t)/
dt
ddt4sin⁡2(t)\frac{d}{d t} 4 \sin^{2}{\left(t \right)}
Transform
Detail solution

  1. The derivative of a constant times a function is the constant times the derivative of the function.

    1. Let u=sin⁡(t)u = \sin{\left(t \right)}.

    2. Apply the power rule: u2u^{2} goes to 2u2 u

    3. Then, apply the chain rule. Multiply by ddtsin⁡(t)\frac{d}{d t} \sin{\left(t \right)}:

      1. The derivative of sine is cosine:

        ddtsin⁡(t)=cos⁡(t)\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}

      The result of the chain rule is:

      2sin⁡(t)cos⁡(t)2 \sin{\left(t \right)} \cos{\left(t \right)}

    So, the result is: 8sin⁡(t)cos⁡(t)8 \sin{\left(t \right)} \cos{\left(t \right)}

  2. Now simplify:

    4sin⁡(2t)4 \sin{\left(2 t \right)}

The answer is:

4sin⁡(2t)4 \sin{\left(2 t \right)}

The graph
02468-8-6-4-2-1010-1010
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
8*cos(t)*sin(t)
8sin⁡(t)cos⁡(t)8 \sin{\left(t \right)} \cos{\left(t \right)}
Simplify
The second derivative [src] 
  /   2         2   \
8*\cos (t) - sin (t)/
8(−sin⁡2(t)+cos⁡2(t))8 \left(- \sin^{2}{\left(t \right)} + \cos^{2}{\left(t \right)}\right)
Simplify
The third derivative [src] 
-32*cos(t)*sin(t)
−32sin⁡(t)cos⁡(t)- 32 \sin{\left(t \right)} \cos{\left(t \right)}
Simplify
The graph
Derivative of 4*sin(t)^(2)
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